diff --git a/src/pyrecest/utils/metrics.py b/src/pyrecest/utils/metrics.py
index 958b802e5..fc9ca392b 100644
--- a/src/pyrecest/utils/metrics.py
+++ b/src/pyrecest/utils/metrics.py
@@ -770,6 +770,19 @@ def _validate_positive_semidefinite(matrix: np.ndarray, name: str) -> None:
 def _as_covariance_matrix(value: ArrayLike, name: str) -> np.ndarray:
     matrix = _as_numeric_array(value, name)
     _validate_square_matrix(matrix, name)
+    if not np.all(np.isfinite(matrix)):
+        raise ValueError(f"{name} must contain only finite values")
+    matrix_transpose = matrix.T
+    matrix_scale = np.maximum(
+        np.maximum(np.abs(matrix), np.abs(matrix_transpose)),
+        1.0,
+    )
+    with np.errstate(over="ignore", under="ignore", invalid="ignore"):
+        relative_asymmetry = np.abs(
+            matrix / matrix_scale - matrix_transpose / matrix_scale
+        )
+    if np.any(relative_asymmetry > 1e-12):
+        raise ValueError(f"{name} must be symmetric")
     matrix = _symmetrize(matrix)
     _validate_positive_semidefinite(matrix, name)
     return matrix
diff --git a/tests/test_metrics_wasserstein_validation.py b/tests/test_metrics_wasserstein_validation.py
index 1307e9d09..7b722318f 100644
--- a/tests/test_metrics_wasserstein_validation.py
+++ b/tests/test_metrics_wasserstein_validation.py
@@ -35,6 +35,25 @@ def test_extent_wasserstein_rejects_indefinite_extents(self):
         ):
             extent_wasserstein_distance(valid, indefinite)
 
+    def test_gaussian_wasserstein_rejects_nonsymmetric_covariances(self):
+        mean = np.zeros(2)
+        valid = np.eye(2)
+        nonsymmetric = np.array([[2.0, 1.0], [0.0, 2.0]])
+
+        with self.assertRaisesRegex(ValueError, "covariance1 must be symmetric"):
+            gaussian_wasserstein_distance(mean, nonsymmetric, mean, valid)
+        with self.assertRaisesRegex(ValueError, "covariance2 must be symmetric"):
+            gaussian_wasserstein_distance(mean, valid, mean, nonsymmetric)
+
+    def test_extent_wasserstein_rejects_nonsymmetric_extents(self):
+        valid = np.eye(2)
+        nonsymmetric = np.array([[2.0, 1.0], [0.0, 2.0]])
+
+        with self.assertRaisesRegex(ValueError, "estimated_extent must be symmetric"):
+            extent_wasserstein_distance(nonsymmetric, valid)
+        with self.assertRaisesRegex(ValueError, "reference_extent must be symmetric"):
+            extent_wasserstein_distance(valid, nonsymmetric)
+
 
 if __name__ == "__main__":
     unittest.main()